Coexistence probability in the last passage percolation model is $6-8\log2$

TL;DR

This paper investigates a competition model on the integer lattice governed by last passage percolation, establishing the probability of coexistence of three clusters and analyzing the density of the central cluster.

Contribution

It proves the exact probability of coexistence of three clusters in the model and links the competition interfaces to multi-TASEP particles, providing new insights into their behavior.

Findings

Coexistence probability is exactly 6 - 8 log 2.

When coexistence occurs, the central cluster has positive density.

The results connect competition interfaces to multi-TASEP particles.

Abstract

A competition model on $\N^{2}$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6-8\log2$. When this happens, we also prove that the central cluster almost surely has a positive density on $\N^{2}$. Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and on recent results about collision in the multi-TASEP.